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Learn the importance of continuity in functions and how it impacts their properties. Explore examples of continuous and non-continuous functions, including polynomials, exponential, logarithmic, sine, cosine, and rational functions. Understand how operations on continuous functions maintain continuity.
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1.7 INTRODUCTION to CONTINUITY
A function is said to be continuouson an interval if the graph has no breaks, jumps, or holes in the interval. Continuity is important because, as we shall see, continuous functions have many desirable properties. For example, for the continuous function f(x) = 3x3 – x2 + 2x -1, since f(0) = -1 and f(1) = 3 we expect there to be a zero between 0 and 1 as shown below.
But notice that for the function f(x) = 1/x, f(-1) = -1 and f(1) = 1 so can we expect there to be a zero between -1 and 1? No, because it is NOT continuous between [-1,1] A continuous function has a graph which can be drawn without lifting the pencil from the paper.
Not continuous in any open interval Containing a positive integer. Exponential functions, logarithmic functions, polynomials, sine, cosine, Rational functions are continuous on any interval in which their denominators are not zero. Functions created by adding, subtracting, multiplying, or composing continuous functions are also continuous
If f(x) is continuous at x = c, the values of f(x) approach f(c) as x approaches c
